Its hard to talk 'efficiency' as they work in different ways.
Some actively 'cool' the air (including through contact with ground or water, or evaporative cooling from plants), some don't further cool the air they cool people through airflow like an electric fan (effectively the human cools themself through evaporation).
Their function will of course also vary through the day and year depending on literally the way the wind blows.
Many also were designed to be attuned to local wind and climate conditions which would have changed over centuries since so its also hard to judge their original function.
Could we have an idea of the temperature change we can expect? For example, if it's 40 degrees celcius outside, can the mechanism go down to 30? 25? 20?
